Derivative of arcsin x: Formula, Proof and Examples

The derivative of \arcsin x is \dfrac{1}{\sqrt{1 - x^2}}. It holds on the open interval -1 < x < 1, where the inverse sine is smooth.

Graph of arcsin x and its derivative

The derivative is never negative, so \arcsin x always rises. Its smallest slope is 1 at the origin, and the curve turns vertical at the endpoints (\pm 1, \pm\frac{\pi}{2}).

Proof of the derivative of arcsin x

Let y = \arcsin x. Then \sin y = x with -\frac{\pi}{2} \le y \le \frac{\pi}{2}. Differentiate both sides with implicit differentiation:

Now write \cos y in terms of x. From \sin^2 y + \cos^2 y = 1 we get \cos y = \pm\sqrt{1 - x^2}. On \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] cosine is never negative, so we take the positive root:

The same idea works for every inverse function, see the inverse function rule.

Chain rule version: derivative of arcsin(u)

With the chain rule, put the inside derivative on top:

The third row is worth remembering. It is why \int \frac{dx}{\sqrt{a^2 - x^2}} = \arcsin\frac{x}{a} + C.

Worked examples

Example 1: a product. Differentiate f(x) = x\arcsin x with the product rule:

Example 2: a slope at a point. The slope of y = \arcsin x at x = \frac{1}{2} is

Example 3: a power of arcsin. For g(x) = (\arcsin x)^2 the outer function is a square:

Common mistakes

1. Reading \sin^{-1} x as \frac{1}{\sin x}. The -1 means inverse function, not reciprocal.
2. Forgetting the inside derivative. \frac{d}{dx}\arcsin(3x) is \frac{3}{\sqrt{1 - 9x^2}}, not \frac{1}{\sqrt{1 - 9x^2}}.
3. Squaring only x in the chain rule. Inside the root you need 1 - u^2, so \arcsin(x^2) gives \sqrt{1 - x^4}, not \sqrt{1 - x^2}.
4. Mixing up the sign with arccos. The derivative of arccos x is the same expression with a minus sign.

Going the other way, the integral of 1/√(1 − x²) is \arcsin x + C, and the integral of arcsin x uses integration by parts. Practise more with inverse trig derivative problems or read about the arcsine function.